In summary, our numerical simulation clearly demonstrates that the ground state of Eq. REF  in one dimension is in an analog of spinon Fermi surface state. Similar to the higher dimension spinon Fermi surface phase, there is Friedel oscillation in the layer polarization \(P_z\) . There should also be a metallic conductance in the counter-flow transport. These results confirm our argument that doping excitons into a layer polarized Mott insulator with spinon Fermi surface naturally results in a phase with neutral Fermi surfaces in both layers. Due to the strong fluctuation in one dimension, magnetic order is not present and a spinon Fermi surface state can be found already in the simple spin model as in Eq. REF . In two dimension, the ground state of Eq. REF  is more likely to be in an exciton superfluid phase described by bosonic parton theory as shown in Sec. . However, in the weak Mott regime, higher order ring exchange couplings needed to be included in Eq. REF . In this case a spinon Fermi surface described by fermionic parton theory may be stabilized[1]}, [2]} and then we expect a phase with neutral Fermi surfaces in both layers at finite \(x\) . We leave the detailed numerical study of the more complicated problem with additional ring exchange couplings to future. Experimentally we note that there is already encouraging signature of spinon Fermi surface at \(x=0\)  limit in MoTe\(_2\) /WSe\(_2\)  system[3]} and we propose to search for smoking gun evidence of the neutral Fermi surface at finite \(x\)  in MoTe\(_2\) -hBN-MoTe\(_2\) /Wse\(_2\)  system from counter-flow transport and Friedel oscillation measurements. Although our numerical simulation in one dimension is mainly used as a guidance for two dimension, the recent experimental progress on one dimensional moiré superlattice[4]} may make the experimental realization of the 1D model also possible in near future.
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